Hyperbolic geometry of Kuramoto oscillator networks

Abstract

Kuramoto oscillator networks have the special property that their\ntrajectories are constrained to lie on the (at most) 3D orbits of the M\\"obius\ngroup acting on the state space $T^N$ (the $N$-fold torus). This result has\nbeen used to explain the existence of the $N-3$ constants of motion discovered\nby Watanabe and Strogatz for Kuramoto oscillator networks. In this work we\ninvestigate geometric consequences of this M\\"obius group action. The dynamics\nof Kuramoto phase models can be further reduced to 2D reduced group orbits,\nwhich have a natural geometry equivalent to the unit disk $\\Delta$ with the\nhyperbolic metric. We show that in this metric the original Kuramoto phase\nmodel (with order parameter $Z_1$ equal to the centroid of the oscillator\nconfiguration of points on the unit circle) is a gradient flow and the model\nwith order parameter $iZ_1$ (corresponding to cosine phase coupling) is a\ncompletely integrable Hamiltonian flow. We give necessary and sufficient\nconditions for general Kuramoto phase models to be gradient or Hamiltonian\nflows in this metric. This allows us to identify several new infinite families\nof hyperbolic gradient or Hamiltonian Kuramoto oscillator networks which\ntherefore have simple dynamics with respect to this geometry. We prove that for\nthe $Z_1$ model, a generic 2D reduced group orbit has a unique fixed point\ncorresponding to the hyperbolic barycenter of the oscillator configuration, and\ntherefore the dynamics are equivalent on different generic reduced group\norbits. This is not always the case for more general hyperbolic gradient or\nHamiltonian flows; the reduced group orbits may have multiple fixed points,\nwhich also may bifurcate as the reduced group orbits vary.\n